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Full text of "Treatise On Analysis Vol-Ii"

3 POSITIVE MEASURES: THE ABSOLUTE VALUE OF A MEASURE 107 By virtue of (13.3.2.1) it is enough to show that |ju(/)| ^ ji(l/l) f°r al* /e JTC(X). Given/, there exists £ e C such that |£| = I and |X/)| = M/) = ; since /* is real, we have MC/) = A<#(f/)) >' and since /j ^ 0 and £ |/|, it follows that /i(#(f/)) ^ Ml/I), which establishes the assertion. If jw is any raz/ measure on X, it follows from (13.3.3) that fi g |ju|, and therefore : (13.3.6) Every real measure on X is the difference of two positive measures (for a more precise result, see (13.15)). If /* is any (complex) measure on X, it follows from (13.3.3) and (13.3.2) that we have (13.3.7) Also if ^, v are any two measures on X we have (13.3.8) |/£ + v| £ |/£| + |v| by virtue of (13.3.2). Finally, it follows immediately from the definitions that, if n : X -» X' is a homeomorphism and ^ is any measure on X, then (13.3.9) brGOl - PROBLEMS 1. Let X be a locally compact space, E a vector subspace of ^R(X) and P a convex cone in ^R(X) (i.e., a subset of this space such that the relations/ep and #ep imply /+ g e P and afe P for all scalars a > 0). Suppose that for each function h e Jf R(X) there exists /e E such that /— h e P, Let if be a real linear form on E such that the relation/e E n P implies u(f) £> 0. Let h e JfR(X) and let Pi be the set of all /e E such that h ~/E P, and P£' the set of all/e E such that/— h e P. Show that these two sets Pi, P£ are nonempty, and that if of is the least upper bound of the «(/) with/e Pi and a" the greatest lower bound of the u(f) with /e Pi', then a' and a" are finite and a" <J a". Deduce that there exists a linear form «i on the subspace E! *» E + Rh of ^R(X) which extends w, such that the relation /i e EI n P implies w^C/i) j> 0. Show that a' g u^(h) <J a" for any such exten- sion «i of u> and that the extension is unique if and only if of = a*.*. is a positive measure on X, then